The American Mathematics Society (AMS) Short Course on Quantum Computation
is in conjunction with the
AMS Annual Meeting to be held in Washington, DC,
January 19-22, 2000.

Synopses and accompanying reading lists follow. Lecture notes will be available to those
who register. Advance Short Course registration fees: $80 ($35/student/unemployed/emeritus); on-site
Short Course registration fees: $95 ($45 student/unemployed/emeritus). Registration for this Short Course
is made through the AMS. One can be registered for the Short Course without being registered for
the AMS Meeting.

The Nobel Laureate Richard Feynman was one of the first
individuals to observe that there is an exponential slow down
when computers based on classical physics, i.e., classical
computers, are used to simulate quantum systems. Richard
Feynman then went on to suggest that it would be far better to
use computers based on quantum mechanical principles, i.e.,
quantum computers, to simulate quantum systems. Such
quantum computers should be exponentially faster than their
classical counterparts.

Interest in quantum computation suddenly exploded when Peter
Shor devised an algorithm for quantum computers that could
factor integers in polynomial time. The fastest known algorithm
for classical computers factors much more slowly, i.e., in
superpolynomial time. Shor's algorithm meant that, if quantum
computers could be built, then cryptographic systems based on
integer factorization, e.g., RSA, could easily be broken. These
cryptographic systems are currently extensively used in banking
and in many other areas. Lov Grover then went on to create a
quantum algorithm that could search data bases faster than any
thing possible on a classical computer. These algorithms are
based on physical principles not implementable on classical
computers, quantum superposition and quantum entanglement.

As a result, the race to build a quantum computer is on. But the
mathematical, physical, and engineering challenges to do so are
formidable, and are a worthy challenge for the best scientific
minds. One of the chief obstacles to creating a quantum
computer is quantum decoherence. By this we mean that
quantum systems want to wander from their computational paths
and quantum entangle with the rest of the environment.

This short course focuses on the mathematical challenges
involved in the development of quantum computers and
quantum algorithms, challenges worthy of the best mathematical
minds. It is hoped that, as a result of this course, many
mathematicians will be enticed into working on the grand
challenge of quantum computation.

The Short Course will begin with an overview of quantum
computation, given in an intuitive and conceptual style. No
prior knowledge of quantum mechanics will be assumed.

In particular, the Short Course will begin with an introduction to
the strange world of the quantum. Such concepts as quantum
superposition, Heisenberg's uncertainty principle, the "collapse"
of the wave function, and quantum entanglement (i.e., EPR
pairs) will be introduced. This will also be interlaced with an
introduction to Dirac notation, Hilbert spaces, unitary
transformations, and quantum measurement.

Some of the topics covered in the course will be:

Quantum teleportation

Shor's quantum factoring algorithm

Grover's algorithm for searching a database

Quantum error-correcting codes

Quantum cryptography

Quantum information theory

Quantum complexity theory, including the quantum
Turing machine

The problems of quantum entanglement and locality

Implementation issues from a mathematical perspective

Each topic will be explained and illustrated with simple
examples.

The course will end with a list of the grand challenges of
quantum computation, i.e., a list of mathematical problems that
must be solved to make the concept of a quantum
computer a reality for the twenty-first century and the
millennium.

A qubit device [1] is a physical implementation of
a set of quantum bits, or qubits, as they are now
commonly known. A qubit [2] is a quantum system with
a two-dimensional Hilbert space, capable of existing
in a superposition of Boolean states, and also capable
of being entangled with the states of other qubits.
The exciting new interdisciplinary fields of quantum
information processing, quantum computing, quantum
communication, and quantum cryptography are rich with
a plethora of potentially useful qubit devices, ranging
from quantum games and quantum teleporters to quantum
copiers and quantum computers [3]. The major obstacle
to the successful development of these devices is
the phenomenon of quantum decoherence, in which even
weak interactions of the qubits with noncomputational
environmental degrees of freedom can destroy the
off-diagonal components of the reduced density matrix
of the qubits, obliterating essential quantum
coherence and quantum entanglement. This lecture presents
brief mathematical descriptions of a variety of
potential qubit devices, alternating with expository
discussions of the issue of quantum decoherence, as
it relates to the possible practical development of
these devices.

The interaction-free detector [4] provides a
simple example of the practical use of path qubits.
(The two-dimensional Hilbert space of a path qubit
represents two possible quantum-interfering
paths of a photon in spacetime.) In this photonic
device, the presence of an opaque object in one arm
of an interferometer destroys the interference of
an incident photon, sometimes signalling the presence
of the object, even though the photon could not have
taken a path intersecting the object. A simple
mathematical analysis of the device is provided.

Another simple example of a photonic qubit device
is a quantum key receiver based on a positive
operator valued measure [5]. This interferometric
device exploits the entanglement of path and
polarization qubits.

The mathematical theory of games is currently being
generalized to include quantum games. To gain some
insight into quantum games, I review a brief
mathematical description of a particularly simple
quantum game involving quantum-coin flipping [6].

In order to develope a multicomponent qubit device,
it is useful to implement various quantum gates.
I provide mathematical descriptions of various photonic
implementations of quantum gates, including the
quantum square-root of not gate, the quantum not
gate, the Hadamard gate, and the quantum controlled-not
gate. A single-photon balanced Mach-Zehnder
interferometer and various photonic qubit entanglers
are also described analytically.

Quantum states are very delicate, so likely some
sort of quantum error correction will be necessary
to build reliable quantum computers. The theory of
quantum error-correcting codes has some close ties
to and some striking differences from the theory of
classical error-correcting codes.
Many quantum codes can be described in terms of the
stabilizer of the codewords. The stabilizer is a
finite Abelian group, and allows a straightforward
characterization of the error-correcting properties
of the code. The stabilizer formalism for quantum
codes also illustrates the relationships to classical
coding theory, particularly classical codes over
GF(4), the finite field with four elements.

This talk will give a survey of quantum topology and topological quantum
field theory, with an eye toward possible interactions with quantum
computing. In the applications in the interface of quantum theory and knot
theory and low dimensional topology there have been numerous models built
that are not quite physical, and yet very informative topologically. Some
of these models impinge on string theory and quantum gravity, yet few of
them involve the unitarity and projection properties so crucial to quantum
computing. In this lecture we shall survey the interface between quantum
topology and quantum computation, with an eye for the possibility of
significant breakthroughs.

Anyons are special particles (more exactly, quasi-particles, or excitations)
in two-dimensional quantum systems. The simplest example of anyons can be
described in terms of stabilizer operators acting on qubits.

Let us consider
a
graph on a surface of genus $g$. The qubits will be associated with the
edges
of this graph. One can define operators
$A_s=\prod_{j\in {\rm star}(s)}\sigma_x^j$
associated to each vertex. Similarly, there are operators
$B_p=\prod_{j\in {\rm boundary}(p)}\sigma_z^j$
associated to the faces. These operators define a quantum code:
the codewords are the vectors which satisfy
$A_s|\xi\rangle=|\xi\rangle$ and $B_p|\xi\rangle=|\xi\rangle$ [1].
Such vectors form a subspace of dimensionality $2^{2g}$.
Changing one of the stabilizer conditions to
$A_s|\xi\rangle=-|\xi\rangle$ or $B_p|\xi\rangle=-|\xi\rangle$
can be interpreted as an ``excitation''. The excitations reveal nontrivial
properties even on the plane: if one moves one excitation around the other,
the quantum state is multiplied by $-1$. Such multiplication by a phase
factor
is a characteristic feature of Abelian anyons.

A generalized version of this model gives rise to nonabelian anyons. Each
qubit can be replaced by a larger quantum system whose basis states are
indexed by elements of any finite group $G$. From the physical point of
view,
the new model is an implementation of a discrete gauge symmetry [2]. One
can
associate a finite-dimensional Hilbert space to each excitation
configuration
on the plane. It is interesting that this space does not have a tensor
product
structure. More exactly, it has a tensor factor associated with each
excitation, but these tensor factors are not protected against errors. The
remaining factor, which is protected, is non-local (i.e. depends on all the
excitations together). One can act on the protected Hilbert space by moving
excitations around each other. Each braid group element (i.e. a
topologically
different way of moving the excitations) is represented by a certain unitary
operator. If one fuses two excitations into one, the Hilbert space
shrinks. Actually, it splits into several Hilbert spaces corresponding to
different types of the new excitation. Thus fusing two excitations is a
measurement. Finilly, if one creates a new pair of excitations, it always
appears in a certain quantum state. All three operations, braiding, fusion,
and creation of a new pair, are intrinsically fault-tolerant due to their
topological nature [3].

An important question about anyons is whether the topological operations
form
a universal computational basis. This turns out to be the case for $G=S_3$,
despite the fact that the image of the braid group in the group of unitary
operators is finite (for any given number of strands). Universality is
achieved in an adaptive manner, i.e. by doing measurements during
computation
and choosing the next braid group generator depending on the previous
measurement outcomes.

It should be noted that anyons are not necessarily related to groups. The
most
general mathematical framework for anyons is a unitary ribbon category.
This
type of an algebraic structure has been studied in connection with braid
group
representations and invariants of knots and 3-manifolds [4].

We explore the many unresolved mathematical problems
associated with quantum entanglement (QE). QE appears to be
that physical phenomenon which gives quantum computing
devices their startling computing power, and which clearly
separates such quantum devices from classical computing
devices.

We begin by noting that the state | \Psi > of a quantum
system can be represented as either an element of a Hilbert space
H (usually finite dimensional for quantum computing devices),
or as a positive definite operator \rho (having trace 1) on H,
called the density operator. We then consider quantum systems
"living" respectively in the Hilbert spaces H_1, ... , H_n. They
are said to be entangled if their joint state | \Psi >, which "lives"
in the tensor product H of H_1, ..., H_n, cannot be factored into
the tensor product of states | \Psi_j > "living" respectively in
H_j, j = 1 ... n. There is a similar definition in terms of density
operators.

After the definition of QE, we discuss the Einstein-Podolsky-Rosen
(EPR) paradox. Then we give two examples of
applications of QE, namely, 1) quantum teleportation and 2) the
Shor quantum factoring algorithm. In each of the two examples,
we draw a contrast between two perspectives, i.e., the
Schrodinger and the Heisenberg models of quantum mechanics.

Next we proceed to study the mathematical structure of QE.
Let SU_j (j=1,2, ... , n) and SU denote respectively the Lie
groups of special unitary transformations on H_j and H. We
define the locality subgroup L(n) of SU as the tensor product of
the n Lie groups SU_j. Two quantum states \rho and \rho' are
said to be locally equivalent if there is a unitary transformation
in L(n) which transforms \rho into \rho'. Under this equivalence
relation, the Lie algebra u of the unitary group U of H
decomposes into L(n)-orbits, called entanglement classes. The
entanglement classes turn out to be symplectic manifolds with a
rich mathematical structure.
We will construct invariants of QE.

Finally, we explore the intriguing possibility of a
relationship between QE and knot theory. We will, for example,
illustrate a relationship between the GHZ state and the
Borromean rings of knot theory.

If time permits, we will give a list of mathematical research
problems.

The model of computation used by today's digital computers is essentially
that proposed by Turing and refined by von Neumann over fifty years ago.
For many years, it was generally assumed that this model was a fundamental
consequence of the mathematics of computation. It now appears that it
instead is a consequence of the laws of physics. It was recently discovered
that hypothetical computers based on quantum mechanical principals can solve
certain problems exponentially faster than the best known algorithms for
solving these problems on classical digital computers. This speedup has not
been proved; in fact, proving such a speedup would solve the famous open
problem of separating the computational complexity classes BPP (probabilistic
polynomial time-bounded computation) and PSPACE (polynomial space-bounded
computation). Quantum computers could also not speed up all computations;
so far, surprisingly few problems have been discovered that can be solved
faster by algorithms on quantum computers.

While based on quantum mechanics, the mathematical model for quantum
computation is straightforward and can be understood without a deep
physics background. The state of the quantum computer is an element
of a tensor product of two-dimensional complex vector spaces, each of which
we call a qubit. The computation is performed by manipulating the
state of the computer via a sequence of unitary transformations; in
each of these transformations, at most two of the qubits generating the
tensor product are allowed to interact. (Replacing two by any larger,
constant, integer generates the same model of computation.) To output
the result, a von Neumann projection measurement is used to extract
information from the state of the computer.

As mentioned above, to date only a very few classes of algorithms have
been discovered where quantum computers provide a substantial speedup. One
class is based on using Fourier transforms to find periodicity. The first
such algorithm was discovered by Dan Simon, and the algorithms for factoring
integers and computing discrete logarithms that I discovered also fall into
this class. A second class includes Lov Grover's algorithm, which speeds up
the time for searching for an item in an unordered list from linear in N
to order of the square root of N, where N is the number of items in the
list. A number of interesting extensions of this algorithm have been
discovered, all using essentially the same techniques. The third class
consists of algorithms for simulating quantum mechanical systems. I plan
to explain the basic model of quantum computation, and show how the first
two classes of algorithms described above work.

Since quantum computation is on the whole a very young field, relatively
few overviews of the area have been written. Two good ones are Alexei
Kitaev's survey article [4] and John Preskill's lecture notes [5]. The
other references listed below are an article proving some basic theorems
about the model of quantum computation [1], and four papers explaining
some of the algorithms described above.

Quantum computation is a fascinating new
area that touches upon the foundations of both quantum
physics and computer science. Quantum computers provide the
first (and only) example of a model of computation that violates
the modern Church-Turing thesis [1],[2],[3]: this thesis states that
any `reasonable' model of computation can be simulated on a
probabilistic Turing machine with at most polynomial factor
simulation overhead.

The complexity class that captures the power of quantum
computers is BQP - the class of languages that can be
recognized (with bounded error probability) in polynomial
time on a quantum Turing Machine. We know that BQP contains
the class BPP of languages that can be recognized in polynomial
time on a probabilistic Turing Machine, and is in turn contained
in the class $P^{\#P}$, of counting problems. One of the most
important open questions in this area is: does BQP contain NP?
An affirmative answer would mean that quantum computers can
efficiently solve many of the most important computational
problems, including the traveling salesman problem. There is
evidence showing that this will be a hard question to resolve -
in much the same way as the P vs. NP problem. It has been proved
that in the black box model, that quantum computers cannot solve
NP-complete problems in subexponential time [4] (indeed even in
o(2^{n/2}) steps).

A study of the relationship between quantum computation,
nondeterminism and interaction (as computational resources)
has already proved to be quite fruitful. It has been recently
shown that BQNP - the quantum analog of the class NP - is
contained in $P^{\#P}$ [5]. There have also been significant
developments in the area of quantum interactive proofs. Here the
issue is the number of rounds of communication
and number of quantum bits that must be exchanged between a
prover and verifier, to convince the (polynomially bounded)
verifier of the answer to a computational problem. It has
been shown that PSPACE has a two round quantum interactive
proof [6].

Quantum communication complexity is another area that has been
extensively studied in the last few years, and have provided
insights into quantum computation and vice-versa. The power of
quantum computation lies in the exponentially many hidden degrees
of freedom in the state of an $n$ quantum bit system --- whereas
$2^n - 1$ complex numbers are necessary to specify the state,
Holevo's theorem states that $n$ quantum bits cannot be used to
communicate any more than $n$ classical bits. Nevertheless, it has
recently been established that there are communication tasks that
can be carried out using exponentially fewer quantum bits than
classical bits [7][8].

In this talk I will give an overview of these new quantum complexity
classes, some of the techniques used in proving the main results in
the area, and a discussion of the open issues in the area. A set of
lecture notes covering some of these topics can be found at the
web site [9].